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Understanding the domain and range of a function is crucial. In this case, we have the function \( f(x) = \sqrt[3]{x} + 1 \). The domain of a function defines all possible input values (x-values) for which the function is defined. For cubic root functions, the cube root is defined for all real numbers. Therefore, the domain of \( f(x) \) is \( (-\infty, \infty) \).

The range of the function represents all possible output values (y-values). Since the cubic root function can produce any real number, adding 1 to this function will still produce any real number. Thus, the range of \( f(x) = \sqrt[3]{x} + 1 \) is also \( (-\infty, \infty) \).

Understanding the domain and range helps in visualizing and graphing the function effectively.

Derivatives are essential for understanding the behavior of a function, such as finding critical points and determining whether the function is increasing or decreasing. To find the derivative of the function \( f(x) = \sqrt[3]{x} + 1 \), we use the power rule.

Recall that \( \sqrt[3]{x} \) can be represented as \( x^{1/3} \). To find the derivative of \( x^{1/3} \), we use the rule: \( \frac{d}{dx} [x^n] = nx^{n-1} \).

Applying this rule, we get:

\[ f'(x) = \frac{1}{3} x^{-2/3} \]
This indicates how the slope of the function \( f(x) \) changes as \( x \) changes.

Since \( f'(x) = \frac{1}{3} x^{-2/3} \) is positive for all real numbers \( x \), the function is always increasing. There are no critical points (points where \( f'(x) = 0 \)), further confirming that the function is steadily increasing.

Graphing functions allows us to visualize their behavior. For the function \( f(x) = \sqrt[3]{x} + 1 \), start by plotting a few key points. Notable points include:

• When \( x = 0 \), then \( f(0) = \sqrt[3]{0} + 1 = 1 \). So, the point \( (0,1) \) is on the graph.
• When \( x = 1 \), then \( f(1) = \sqrt[3]{1} + 1 = 2 \). So, the point \( (1,2) \) is also on the graph.
• When \( x = -1 \), then \( f(-1) = \sqrt[3]{-1} + 1 = 0 \). This makes \( (-1,0) \) another point on the graph.

By plotting these points, you will see that the function consistently increases. The cubic root function itself has a characteristic shape, and shifting it upward by 1 unit retains this shape but moves it up.

This visual understanding, combined with our knowledge of the function's domain and range, allows us to effectively graph and interpret \( f(x) = \sqrt[3]{x} + 1 \).